Properties

Label 3380.2.a.k
Level $3380$
Weight $2$
Character orbit 3380.a
Self dual yes
Analytic conductor $26.989$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9894358832\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} - q^{5} -\beta_{1} q^{7} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} - q^{5} -\beta_{1} q^{7} + ( -1 + \beta_{2} ) q^{9} + ( 1 + \beta_{1} - \beta_{2} ) q^{11} + \beta_{1} q^{15} + ( -2 + 3 \beta_{1} - 2 \beta_{2} ) q^{17} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{19} + ( 2 + \beta_{2} ) q^{21} + ( -1 + \beta_{2} ) q^{23} + q^{25} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{27} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{29} + ( -4 + 3 \beta_{1} - 2 \beta_{2} ) q^{31} + ( -1 - \beta_{1} ) q^{33} + \beta_{1} q^{35} + 3 q^{37} + ( 1 + 4 \beta_{1} - \beta_{2} ) q^{41} + ( -3 + 5 \beta_{1} - 3 \beta_{2} ) q^{43} + ( 1 - \beta_{2} ) q^{45} + ( 7 - 4 \beta_{1} + 7 \beta_{2} ) q^{47} + ( -5 + \beta_{2} ) q^{49} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{51} + ( -\beta_{1} + 7 \beta_{2} ) q^{53} + ( -1 - \beta_{1} + \beta_{2} ) q^{55} + ( -\beta_{1} - \beta_{2} ) q^{57} + ( -5 - 4 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -9 - \beta_{1} + \beta_{2} ) q^{61} + ( -1 + \beta_{1} - \beta_{2} ) q^{63} + ( 3 + 4 \beta_{1} - 4 \beta_{2} ) q^{67} + ( -1 + \beta_{1} - \beta_{2} ) q^{69} + ( -2 - 3 \beta_{2} ) q^{71} + ( 6 + 2 \beta_{1} + 4 \beta_{2} ) q^{73} -\beta_{1} q^{75} + ( -1 - \beta_{1} ) q^{77} + ( -5 + 5 \beta_{1} + 3 \beta_{2} ) q^{79} + ( -4 + \beta_{1} - 6 \beta_{2} ) q^{81} + ( -6 + 4 \beta_{1} + 3 \beta_{2} ) q^{83} + ( 2 - 3 \beta_{1} + 2 \beta_{2} ) q^{85} + ( 4 + \beta_{1} + 3 \beta_{2} ) q^{87} + ( 5 - 8 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -4 + 4 \beta_{1} - \beta_{2} ) q^{93} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{95} + ( -1 + \beta_{1} - 8 \beta_{2} ) q^{97} + ( -1 - 2 \beta_{1} + 4 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{3} - 3q^{5} - q^{7} - 4q^{9} + O(q^{10}) \) \( 3q - q^{3} - 3q^{5} - q^{7} - 4q^{9} + 5q^{11} + q^{15} - q^{17} + 5q^{21} - 4q^{23} + 3q^{25} + 2q^{27} - 2q^{29} - 7q^{31} - 4q^{33} + q^{35} + 9q^{37} + 8q^{41} - q^{43} + 4q^{45} + 10q^{47} - 16q^{49} - 9q^{51} - 8q^{53} - 5q^{55} - 17q^{59} - 29q^{61} - q^{63} + 17q^{67} - q^{69} - 3q^{71} + 16q^{73} - q^{75} - 4q^{77} - 13q^{79} - 5q^{81} - 17q^{83} + q^{85} + 10q^{87} + 9q^{89} - 7q^{93} + 6q^{97} - 9q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.80194
0.445042
−1.24698
0 −1.80194 0 −1.00000 0 −1.80194 0 0.246980 0
1.2 0 −0.445042 0 −1.00000 0 −0.445042 0 −2.80194 0
1.3 0 1.24698 0 −1.00000 0 1.24698 0 −1.44504 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3380.2.a.k 3
13.b even 2 1 3380.2.a.l yes 3
13.d odd 4 2 3380.2.f.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3380.2.a.k 3 1.a even 1 1 trivial
3380.2.a.l yes 3 13.b even 2 1
3380.2.f.g 6 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3380))\):

\( T_{3}^{3} + T_{3}^{2} - 2 T_{3} - 1 \)
\( T_{7}^{3} + T_{7}^{2} - 2 T_{7} - 1 \)
\( T_{19}^{3} - 7 T_{19} + 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -1 - 2 T + T^{2} + T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( -1 - 2 T + T^{2} + T^{3} \)
$11$ \( -1 + 6 T - 5 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( 13 - 16 T + T^{2} + T^{3} \)
$19$ \( 7 - 7 T + T^{3} \)
$23$ \( -1 + 3 T + 4 T^{2} + T^{3} \)
$29$ \( 13 - 15 T + 2 T^{2} + T^{3} \)
$31$ \( -7 + 7 T^{2} + T^{3} \)
$37$ \( ( -3 + T )^{3} \)
$41$ \( 113 - 9 T - 8 T^{2} + T^{3} \)
$43$ \( 83 - 44 T + T^{2} + T^{3} \)
$47$ \( 559 - 53 T - 10 T^{2} + T^{3} \)
$53$ \( -169 - 79 T + 8 T^{2} + T^{3} \)
$59$ \( -41 + 31 T + 17 T^{2} + T^{3} \)
$61$ \( 881 + 278 T + 29 T^{2} + T^{3} \)
$67$ \( 13 + 59 T - 17 T^{2} + T^{3} \)
$71$ \( -13 - 18 T + 3 T^{2} + T^{3} \)
$73$ \( 8 + 20 T - 16 T^{2} + T^{3} \)
$79$ \( -797 - 58 T + 13 T^{2} + T^{3} \)
$83$ \( -587 + 10 T + 17 T^{2} + T^{3} \)
$89$ \( 953 - 169 T - 9 T^{2} + T^{3} \)
$97$ \( 167 - 121 T - 6 T^{2} + T^{3} \)
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